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Cesàro summable difference sequence space
Journal of Inequalities and Applications volume 2013, Article number: 315 (2013)
The difference sequence spaces , and were introduced by Kizmaz (Can. Math. Bull. 24:169-176, 1981). In this paper, we introduce the Cesáro summable difference sequence space which strictly includes the spaces and but overlaps with . It is shown that the newly introduced space turns out to be an inseparable BK space which does not possess any of the following: AK property, monotonicity, normality and perfectness. The Köthe-Toeplitz duals of are computed and as an application, the matrix classes , and are also characterized.
MSC:40C05, 40A05, 46A45.
1 Notations and definitions
By s we shall denote the linear space of all complex sequences over ℂ (the field of complex numbers). , c and denote the spaces of all bounded, convergent and null sequences with complex terms, respectively, normed by .
Throughout this paper, unless otherwise specified, we write for and for .
A complete metric linear space is called a Frèchet space. Let X be a linear subspace of s such that X is a Frèchet space with continuous coordinate projections. Then we say that X is an FK space. If the metric of an FK space is given by a complete norm, then we say that X is a BK space.
We say that an FK space X has AK, or has the AK property, if , the sequence of unit vectors, is a Schauder basis for X.
A sequence space X is called
normal (or solid) if whenever , , for some ,
monotone if it contains the canonical preimages of all its stepspaces,
sequence algebra if whenever ,
convergence free when, if is in X and if whenever , then is in X.
The idea of dual sequence spaces was introduced by Köthe and Toeplitz  whose main results concerned α-duals; the α-dual of being defined as
Obviously, , where ϕ is the well-known sequence space of finitely non-zero scalar sequences. Also, if , then for . For any sequence space X, we denote by , where . It is clear that , where .
For a sequence space X, if then X is called a Köthe space or a perfect sequence space.
A sequence space of complex numbers is said to be summable (or Cesàro summable of order 1) to if , where . By we shall denote the linear space of all summable sequences of complex numbers over ℂ, i.e.,
It is easy to see that is a BK space normed by
The notion of difference sequence space was introduced by Kizmaz  in 1981 as follows:
2 Motivation and introduction
During the last 32 years, a large amount of work has been carried out by many mathematicians regarding various generalizations of difference sequence spaces of Kizmaz. Keeping aside some exceptions (see, for instance, [7, 8]), in most of these works, the underlying spaces have remained the same, i.e., , c and . In the present work, we take the opportunity to introduce a difference sequence space with underlying space as .
We observe that
as but ,
as but , and
Thus the sequence spaces and overlap but do not contain each other. Similarly, and also overlap without containing each other as is clear from the fact that , and . Note that the sequence is summable but not bounded, whereas the sequence given by , and
is bounded but not summable. This has motivated the authors to look for a new sequence space which properly includes the spaces , and .
We now introduce a sequence space , Cesàro summable difference sequence space, as follows:
The overall picture regarding inclusions among the already existing spaces , c, , , , , and the newly introduced space is as shown below:
In this paper we show that strictly includes the spaces and but overlaps with . It is shown that the newly introduced space is an inseparable BK space which does not possess any of the following: AK property, monotonicity, normality and perfectness. The Köthe-Toeplitz duals of are computed, and as an application, the matrix classes , and are also characterized.
3 Inclusion theorems and topological properties of
We begin with elementary inclusion theorems justifying that is much wider than , and .
Theorem 3.1 , the inclusion being strict.
Proof Let . Then there exists such that for all , and so as . For strict inclusion, observe that but . □
Theorem 3.2 , the inclusion being strict.
Proof For , we have , and so as . Inclusion is strict in view of the example cited in Theorem 3.1. □
Theorem 3.3 , the inclusion being strict.
Proof Inclusion is obvious since . To see that the inclusion is strict, consider the sequence . □
Remark 3.4 Let X and Y be sequence spaces. If , then .
Proof Since , there is a sequence such that . Consider the sequence . Then but . □
Remark 3.5 We have already observed that and , so by Remark 3.4, it follows that neither nor . Also, we have . In view of this and Theorem 3.3, we can say that strictly includes and hence but overlaps with .
We now study the linear topological structure of .
Theorem 3.6 is a BK space normed by
The proof is a routine verification by using ‘standard’ techniques and hence is omitted.
Theorem 3.7 is not separable.
Proof Let A be the set of all sequences , where
with ; . Clearly, and A is uncountable. Let D be any dense set in .
Define a map as follows:
Let . As D is dense in , so there exists some such that .
We set .
For , we have
and already we have , therefore . Hence f is one-to-one. As so D is uncountable. Thus, has no countable dense set. □
Corollary 3.8 does not have a Schauder basis.
The result follows from the fact that if a normed space has a Schauder basis, then it is separable.
Corollary 3.9 does not have the AK property.
Theorem 3.10 is not normal (solid) and hence neither perfect nor convergence free.
Proof Taking and , we see that but although , and so is not normal. It is well known  that every perfect space, and also every convergence free space, is normal and consequently is neither perfect nor convergence free. □
Theorem 3.11 is neither monotone nor a sequence algebra.
Proof Take . Consider where
i.e., . Then and so , i.e., and hence is not monotone. To see that is not a sequence algebra, take and observe that but . □
4 Köthe-Toeplitz duals of
In this section we compute the Köthe-Toeplitz duals of and show that is not perfect.
Proof Let . For any , we have , i.e., and so there exists some such that for and hence , which implies that
Conversely, let . Then for all . Taking for all , we have whence . □
Proof Taking and in [, Theorem 2.13], we have and the result follows in view of Remark 4.2. □
Corollary 4.4 is not perfect.
The proof follows at once when we observe that the sequence but does not belong to .
Proof Let and . Then . For , we have
Obviously, and . We define by and for all . Then and since , the series converges absolutely.
Conversely, if , then converges for all . In particular, taking for all k, we have converges and so converges for all . Since if and only if , we have . □
Corollary 4.6 .
5 Matrix maps
Finally, we characterize certain matrix classes. For any complex infinite matrix , we shall write for the sequence in the n th row of A. If X, Y are any two sets of sequences, we denote by the class of all those infinite matrices such that the series converges for all () and the sequence is in Y for all .
The following theorem is well known.
Theorem 5.1 [, p.219] Let X and Y be BK spaces and suppose that is an infinite matrix such that for each , i.e., , then is a bounded linear operator.
Theorem 5.2 if and only if .
Proof Suppose that and . Proceeding as in Theorem 4.5, we have .
which yields the absolute convergence of for each , and finally we have
for all .
Conversely, by Theorem 5.1, we have
for each and .
Choose any and any and define
Then with . Inserting this value of in (5.1) , we have
Letting and noting that (5.2) holds for every , we are through. □
Remark 5.3 If , then there exists some such that . We shall call ℓ the limit of the sequence and by we shall denote that subset of for which limits are preserved.
Theorem 5.4 if and only if
for each k,
Proof Let the conditions (i)-(iv) hold and suppose that with . It is implicit in (i) that, for each , converges. It follows that converges, whence
Let , and .
Then, for any , we have
Letting , we have as . Making use of this and also of (ii) and (iv) in (5.3), we get the result.
Conversely, let . Then for all . By the same argument as in Theorem 5.2, we have . Taking , we get with for each k. Also, for , we have with , and finally yields . □
Theorem 5.5 if and only if
for each k,
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The authors are grateful to the referee for his/her valuable comments and suggestions, which have improved the presentation of the paper.
The authors declare that they have no competing interests.
VKB and SG contributed equally. All authors read and approved the final manuscript.
An erratum to this article is available at http://dx.doi.org/10.1186/1029-242X-2014-11.
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Cite this article
Bhardwaj, V.K., Gupta, S. Cesàro summable difference sequence space. J Inequal Appl 2013, 315 (2013). https://doi.org/10.1186/1029-242X-2013-315
- sequence space
- BK space
- Schauder basis
- Köthe-Toeplitz duals
- matrix map